This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Okay, sucess, let's break this down like we're drawing a picture! Imagine you're drawing a line on a piece of paper without lifting your pencil. If you have to lift your pencil because there's a hole or a break in your drawing, that's a point of discontinuity. For math problems with fractions, these breaks usually happen when the bottom part of the fraction (the denominator) becomes zero, because you can't divide by zero! Let's find where each of these functions "breaks": 1) Function (i): f(z) = (z+z+0)/(z+7) Step 1: Simplify the function. f(z) = (2z)/(z+7) Step 2: Find where the bottom part is zero. z+7 = 0 z = -7 The point of discontinuity* for f(z) is z = -7. 2) Function (ii): g(z) = (z²-4)/(z-1) Step 1: Find where the bottom part is zero. z-1 = 0 z = 1 The point of discontinuity* for g(z) is z = 1. 3) Function (iii): h(z) = (z+2)(z+1)/(z+2) Step 1: Find where the bottom part is zero. z+2 = 0 z = -2 Even though you could simplify this function by canceling out (z+2) from the top and bottom (making it h(z) = z+1), the original function is still undefined at z = -2 because it would make the denominator zero. So, there's a "hole" in the graph at that point. The point of discontinuity* for h(z) is z = -2. 3 done, 2 left today. You're making progress.